Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

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dc.creatorGraham, R. L.
dc.creatorLagarias, J. C.
dc.creatorMallows, C. L.
dc.creatorWilks, A. R.
dc.creatorYan, C. H.
dc.date2000-10-30
dc.date2005-03-11
dc.date.accessioned2026-07-07T06:25:31Z
dc.date.available2026-07-07T06:25:31Z
dc.descriptionApollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)$\times$(center) is an integer vector. This series of papers explain such properties. A {\em Descartes configuration} is a set of four mutually tangent circles with disjoint interiors. We describe the space of all Descartes configurations using a coordinate system $\sM_\DD$ consisting of those $4 \times 4$ real matrices $\bW$ with $\bW^T \bQ_{D} \bW = \bQ_{W}$ where $\bQ_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and $\bQ_W$ of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the parameter space $\sM_\DD$. We observe that the Descartes configurations in each Apollonian packing form an orbit under a certain finitely generated discrete group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups, the dual Apollonian group and the super-Apollonian group, which have nice geometrically interpretations. We show these groups are hyperbolic Coxeter groups.
dc.description42 pages, 11 figures. Extensively revised version on June 14, 2004. Revised Appendix B and a few changes on July, 2004. Slight revision on March 10, 2005
dc.identifierhttps://arxiv.org/abs/math/0010298
dc.identifierhttp://arxiv.org/abs/math/0010298
dc.identifierDiscrete and Computational Geometry, 34: 547-585(2005)
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/96849
dc.subjectMetric Geometry
dc.subjectGroup Theory
dc.subjectNumber Theory
dc.titleApollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group
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